# Flat Littlewood Polynomials Exist

## TL;DR

This paper proves the existence of Littlewood polynomials with coefficients ±1 that are uniformly bounded and bounded away from zero on the unit circle, confirming a long-standing conjecture from 1966.

## Contribution

It establishes the existence of flat Littlewood polynomials with bounded magnitude on the unit circle, resolving a conjecture by Littlewood from 1966.

## Key findings

- Existence of Littlewood polynomials with bounded magnitude on the unit circle.
- Constants elta, elta exist such that elta \u221a n ndelta nd or all degrees n.
- Confirms a long-standing conjecture in polynomial analysis.

## Abstract

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant \Delta\sqrt{n}$$ for all $z\in\mathbb{C}$ with $|z|=1$. This confirms a conjecture of Littlewood from 1966.